Question

**#1.) in the diagram below $overline{nl}$ bisects $angle mnk$. the $mangle knm=(x^{2}-9)^{circ}$ and the $mangle lnk=(4x)^{circ}$. determine the $mangle lnm$.
a. $9^{circ}$
b. $36^{circ}$
c. $72^{circ}$
d. $4^{circ}$
#2.) $overrightarrow{km}$ bisects $angle jkn$. which of the following below is true?
a) $mangle mkn = mangle lkn$
b) $mangle jkm = mangle mkn$
c) $2(mangle jkn)=mangle mkn$
d) $\frac{1}{2}(mangle jkm)=mangle jkn$
self - assessment: how would you rate your understanding of todays lesson?
what question(s) do you still have about this lesson?

Explanation:

Step1: Use angle - bisector property

Since $\overline{NL}$ bisects $\angle MNK$, then $m\angle KNM = 2m\angle LNK$. So we set up the equation $x^{2}-9 = 2(4x)$.

Step2: Rearrange the equation to quadratic form

$x^{2}-9 = 8x$, which can be rewritten as $x^{2}-8x - 9=0$.

Step3: Factor the quadratic equation

We factor $x^{2}-8x - 9$ into $(x - 9)(x+ 1)=0$.

Step4: Solve for x

Setting each factor equal to zero gives $x - 9=0$ or $x + 1=0$. So $x=9$ or $x=-1$. Since an angle measure cannot give a negative value when substituting back into the angle - expressions, we use $x = 9$.

Step5: Find $m\angle LNM$

Since $m\angle LNM=m\angle LNK$ (because of angle - bisector) and $m\angle LNK=(4x)^{\circ}$, substituting $x = 9$ gives $m\angle LNK=4\times9=36^{\circ}$, so $m\angle LNM = 36^{\circ}$.

For question #2:
A bisector divides an angle into two equal - measure angles. Since $\overrightarrow{KM}$ bisects $\angle JKN$, then $m\angle JKM=m\angle MKN$.

Answer:

1. B. $36^{\circ}$
2. B. $m\angle JKM = m\angle MKN$